Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four

This undergraduate thesis is focused on introducing the reader to concepts related to the search for topological obstructions to the existence of compact gradient shrinking Ricci soliton metrics in dimension four.

It contains a discussion of the relevant background material for this subject.

Furthermore, it introduces the problem of extending the Hitchin-Thorpe inequality to gradient shrinking Ricci soliton metrics and explores the limitations of current results in that direction.

At last, the topic of compact Kaehler gradient shrinking Ricci solitons is introduced and the classification of these spaces is outlined in literature-study fashion.